L(s) = 1 | + 10·5-s + 98·7-s − 54·11-s + 240·13-s − 1.90e3·17-s − 400·19-s − 2.93e3·23-s − 1.89e3·25-s − 5.54e3·29-s + 4.90e3·31-s + 980·35-s + 2.95e3·37-s − 6.07e3·41-s + 3.30e3·43-s − 1.69e4·47-s + 7.20e3·49-s − 6.89e3·53-s − 540·55-s − 5.38e4·59-s + 4.14e3·61-s + 2.40e3·65-s + 3.35e4·67-s − 7.35e4·71-s + 128·73-s − 5.29e3·77-s + 5.77e4·79-s − 2.30e4·83-s + ⋯ |
L(s) = 1 | + 0.178·5-s + 0.755·7-s − 0.134·11-s + 0.393·13-s − 1.59·17-s − 0.254·19-s − 1.15·23-s − 0.606·25-s − 1.22·29-s + 0.916·31-s + 0.135·35-s + 0.354·37-s − 0.563·41-s + 0.272·43-s − 1.12·47-s + 3/7·49-s − 0.337·53-s − 0.0240·55-s − 2.01·59-s + 0.142·61-s + 0.0704·65-s + 0.912·67-s − 1.73·71-s + 0.00281·73-s − 0.101·77-s + 1.04·79-s − 0.366·83-s + ⋯ |
Λ(s)=(=(254016s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(254016s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
254016
= 26⋅34⋅72
|
Sign: |
1
|
Analytic conductor: |
6534.04 |
Root analytic conductor: |
8.99074 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 254016, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
= |
0 |
L(21) |
= |
0 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
| 7 | C1 | (1−p2T)2 |
good | 5 | D4 | 1−2pT+1994T2−2p6T3+p10T4 |
| 11 | D4 | 1+54T+284302T2+54p5T3+p10T4 |
| 13 | D4 | 1−240T+739862T2−240p5T3+p10T4 |
| 17 | D4 | 1+1906T+1860002T2+1906p5T3+p10T4 |
| 19 | D4 | 1+400T+3279798T2+400p5T3+p10T4 |
| 23 | D4 | 1+2930T+9774686T2+2930p5T3+p10T4 |
| 29 | D4 | 1+5540T+40407182T2+5540p5T3+p10T4 |
| 31 | D4 | 1−4904T+59914302T2−4904p5T3+p10T4 |
| 37 | D4 | 1−2952T+138794486T2−2952p5T3+p10T4 |
| 41 | D4 | 1+6070T+232254602T2+6070p5T3+p10T4 |
| 43 | D4 | 1−3304T+197837766T2−3304p5T3+p10T4 |
| 47 | D4 | 1+16988T+429309854T2+16988p5T3+p10T4 |
| 53 | D4 | 1+6896T+613869254T2+6896p5T3+p10T4 |
| 59 | D4 | 1+53820T+1699029142T2+53820p5T3+p10T4 |
| 61 | D4 | 1−68pT+1287381294T2−68p6T3+p10T4 |
| 67 | D4 | 1−33516T+2261444678T2−33516p5T3+p10T4 |
| 71 | D4 | 1+73518T+4934300734T2+73518p5T3+p10T4 |
| 73 | D4 | 1−128T−360358674T2−128p5T3+p10T4 |
| 79 | D4 | 1−57740T+6120687198T2−57740p5T3+p10T4 |
| 83 | D4 | 1+23016T−4303421546T2+23016p5T3+p10T4 |
| 89 | D4 | 1−141530T+14809201898T2−141530p5T3+p10T4 |
| 97 | D4 | 1+226216T+29691907182T2+226216p5T3+p10T4 |
show more | | |
show less | | |
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.698621357691645855154396779744, −9.632426649691190045388261930508, −8.924595907220051428186947398244, −8.605047807448879723279010070588, −7.999466963441445661836865760849, −7.85000536979697586208624090541, −7.19640931536114569642355592199, −6.61340208993397008404142752301, −6.13989783238775839149158436717, −5.88644039374264784313795615549, −4.95751104261027156037606843425, −4.84544816534123082318780079736, −3.97798755369182886633256897094, −3.85924955071786598137556514679, −2.83323491538201949059364176099, −2.30328644003832381056158689407, −1.72523941582283573050028917718, −1.26019755765368447561610254117, 0, 0,
1.26019755765368447561610254117, 1.72523941582283573050028917718, 2.30328644003832381056158689407, 2.83323491538201949059364176099, 3.85924955071786598137556514679, 3.97798755369182886633256897094, 4.84544816534123082318780079736, 4.95751104261027156037606843425, 5.88644039374264784313795615549, 6.13989783238775839149158436717, 6.61340208993397008404142752301, 7.19640931536114569642355592199, 7.85000536979697586208624090541, 7.999466963441445661836865760849, 8.605047807448879723279010070588, 8.924595907220051428186947398244, 9.632426649691190045388261930508, 9.698621357691645855154396779744